where \(M_{\rm eq} = 2.8 \times 10^{17}\,M_\odot \)  is the horizon mass at matter-radiation equality. The numerical factor is \(2( 1 + \Omega _b / \Omega _{\rm CDM} )\) , with \(\Omega _{\rm CDM} = 0.245\)  and \(\Omega _b = 0.0456\)  being the CDM and baryon density parameters [1]}. We will often call \(f_{\rm PBH}( M )\)  mass function in the following. By integrating \(f_{\rm PBH}( M )\)  over its full support we get the total fraction of PBH:
\(f_{\rm PBH}^{\rm tot}\equiv \int \,d\ln M \,f_{\rm PBH}( M ).\) 
